E. KIRKMAN, J. KUZMANOVICH AND J.J. ZHANG

Transcription

1 NAKAYAMA AUTOMORPHISM AND RIGIDITY OF DUAL REFLECTION GROUP COACTIONS arxiv: v1 [math.ra] 6 May 2016 E. KIRKMAN, J. KUZMANOVICH AND J.J. ZHANG Abstract. We study homological properties and rigidity of group coactions on Artin-Schelter regular algebras. 0. Introduction The classical Shephard-Todd-Chevalley Theorem states that if G is a finite group acting faithfully on a finite dimensional C-vector space n i=1 Cx i, then the fixed subring C[x 1,,x n ] G is isomorphic to C[x 1,,x n ] if and only if G is generated by pseudo-reflections of n i=1 Cx i. Such a group G is called a reflection group. The indecomposable complex reflection groups are classified by Shephard and Todd [ST], and there are three infinite families plus 34 exceptional groups. When the commutative polynomial rings are replaced by skew polynomial rings, one can define a notion of reflection group in this noncommutative setting. Then there is one extra class of mystic reflection groups M(a,b,c), discovered in [KKZ2], also see [BB] for further discussion. When G is replaced by a semisimple Hopf algebra H (which is not a group algebra), then there is noinner faithful action ofh on the commutativepolynomialring by a very nice result of Etingof-Walton [EW1]. This is one of many reasons why we need to consider noncommutative Artin-Schelter regular algebras if we want to have a fruitful noncommutative invariant theory. Artin-Schelter regular algebras were introduced by Artin-Schelter [AS] in 1980 s, and by now, are considered a natural analogue of commutative polynomial rings in many respects. The definition of an Artin-Schelter regular algebra (abbreviated by AS regular) is given in Definition 1.1. Throughout the rest of this paper, let k be a base field of characteristic zero, and all vector spaces, (co)algebras, and morphisms are over k. Let H be a semisimple Hopf algebra and let K be the k-linear dual of H. Then K is also a semisimple Hopf algebra. It is well-known that a left H-action on an algebra A is equivalent to a right K-coaction on A, and we will use this fact freely. In this paper we are interested in the case when H is k G := Hom k (kg,k), or equivalently, K is the group algebra kg for some finite group G. Let e denote the unit of G. As an algebra, k G = g G k p g where its multiplication is determined by p g p h = { p g g = h, 0 g h, 1 = p g, g G 2000 Mathematics Subject Classification. 16E10, 16E65, 16W30, 20J50. Key words and phrases. Artin-Schelter regular algebra, dual reflection group, Frobenius algebra, Hilbert series, fixed subring, Nakayama automorphism, rigidity. 1

2 2 E. KIRKMAN, J. KUZMANOVICH AND J.J. ZHANG and its coalgebra structure is determined by for all g G. (p g ) = h Gp h p h 1 g, ǫ(p g ) = δ e,g := { 1 g = e, 0 g e, Definition 0.1. A finite group G is called a dual reflection group if the Hopf algebra H := k G acts homogeneously and inner faithfully on a noetherian AS regular domain A generated in degree 1 such that the fixed subring A H is again AS regular, i.e. the identity component of A under the G-grading is AS regular. In this case we say that G coacts on A as a dual reflection group. An example of a dual reflection group acting on an AS regular algebras is given in Example 3.7; further examples will appear in [KKZ5]. We do not yet have a classification of dual reflection groups. When we have a Hopf algebra (co)action on an AS regular algebra, the homological (co)determinant is defined in [KKZ3]. Definition 0.2. Suppose a finite group G coacts on A as a dual reflection group. Let D be the homological codeterminant of the kg-coaction on A as defined in [KKZ3, Definition 6.2] or [CWZ, Definition 1.4(b)]. Then we call the element m := D 1 in G the mass element of the kg-coaction (or k G -action) on A. Our first result is the following. Let H A (t) be the Hilbert series of a graded algebra A. Theorem 0.3. Let A be a noetherian AS regular domain generated in degree 1. Let G coact on A inner faithfully as a dual reflection group and H = k G. Then the following hold. (1) There is a set of homogeneous elements {f g g G} A with f e = 1 such that A = g G A g and A g = f g A H = A H f g for all g G. (2) There is a generating subset R of the group G satisfying e R such that A 1 = g R kf g (A 1 A H ). (3) Let l R (g) denote the (reduced) length of g G with respect to R. Then degf g = l R (g) for all g G. (4) The mass element m is the unique element in G of the maximal length with respect to l R defined as in part (3). (5) Let p(t) = H A (t)h A H(t) 1. Then p(t) is a product of cyclotomic polynomials, p(1) = G and degp(t) = l R (m). Bypart(4)oftheabovetheorem, misalsothemalth(=maximallength)element in G. Hence a purely homologically defined invariant m has strong combinatorial flavor. Indeed this mass element will appear in several other results. The Nakayama automorphism of an AS regular (or AS Gorenstein) algebra is an important invariant of the algebra. In the study of noetherian Hopf algebras, the explicit expression of the Nakayama automorphism has several applications in Poincaré duality [BZ, Corollary 0.4], Radford s S 4 formula [BZ, Theorem 0.6], and so on. The result about the Nakayama automorphism of a smash product [RRZ2, Theorem 0.2] partially recovers a number of previous results concerning Calabi- Yau algebras; see the discussion in [RRZ2, Introduction]. In [RRZ3], a further connection between the Nakayama automorphism of an AS regular algebra and the

3 DUAL REFLECTION GROUPS 3 Nakayama automorphism of its Ext-algebra is studied. In [LMZ], the Nakayama automorphism has been used to study the automorphism group and a cancellation problem of algebras. The Nakayama automorphism is also essential in the study of rigid dualizing complexes [BZ, VdB]. The main purpose of this paper is to describe the Nakayama automorphism and to study the homological properties of various algebras related to the dual reflection group coaction on AS regular algebras. We will recall several definitions in Section 1. Let A be any algebra. We use µ A to denote the Nakayama automorphism of A. If A is a connected graded algebra, then the graded Nakayama automorphism µ A is unique (if exists) since there is no nontrivial invertible homogeneous element in A. When A is a connected graded noetherian AS Gorenstein algebra, we usually require µ A to be the graded Nakayama automorphism of A. From now on, let (f g,r,m,p(t)) be as in Theorem 0.3. In the setting oftheorem 0.3, we define the covariant ring of the H-action on A to be A cov H := A/((A H ) 1 ) where ((A H ) 1 ) is the ideal of A generated by homogeneous invariant elements in A H ofpositive degree. The covariantring is also called the coinvariantring by some authors. Theorem 0.4. Under the hypothesis of Theorem 0.3. (1) The covariant ring A cov H is Frobenius of dimension G. (2) The Hilbert series of A cov H is p(t). As a consequence, the coefficients of p(t) are positive and the coefficient of the leading term of p(t) is 1. (3) Let f g be the homomorphic image of f g in A cov H for each g G. Then {f g g G} is a k-linear basis of A cov H and the graded Nakayama automorphism of A cov H is of the form µ A cov H : f g β(g)f mgm 1, for all g G, where {β(g)} g G are some nonzero scalars in k. (4) The scalars in part (3) satisfy β(m) = 1, and if m commutes with g and h and l R (gh) = l R (g)+l R (h) then β(gh) = β(g)β(h). As in Theorems 0.3 and 0.4, the mass element m has special properties. We can say more next. Theorem 0.5. Under the hypothesis of Theorem 0.3. (1) f m is a homogeneous normal element in A. (2) Let φ m be the conjugation automorphism of A defined by a f m af 1 m, for all a A. Then φ m : f g β(g)f mgm 1, for all g G, where β(g) are defined as in Theorem 0.4(3). In other words, the graded Nakayama automorphism of the covariant ring A cov H is induced by the conjugation automorphism φ m. (3) Any N-graded algebra automorphism of A cov H commutes with µ A cov H.

4 4 E. KIRKMAN, J. KUZMANOVICH AND J.J. ZHANG (4) The automorphism φ m sends A H to A H. The next result tells us about the Nakayama automorphisms of other algebras related to A. The homological determinant of a Hopf algebra action is defined in [KKZ3, Definition 3.3]. Theorem 0.6. Under the hypothesis of Theorem 0.3. Let η m = φ 1 m. (1) µ A H = (η m µ A ) A H. As a consequence, µ A sends A H to A H. (2) Three automorphisms µ A H, µ A A H and η m A H commute with each other. (3) The homological determinant of the H-action on A is the projection from H k p m 1. (4) Let trans l m be the left translation automorphism of H = k G defined by trans l m : p g p mg, for all g G. Then the Nakayama automorphism of A#H is equal to µ A #trans l m. Classically, when A is a commutative polynomial ring and G is a reflection group, the covariant ring is clearly a complete intersection, so the following is a natural question. Question 0.7. Under the hypothesis of Theorem 0.3 (or more generally for a semisimple Hopf algebra H), is the covariant ring A cov H a complete intersection of GK type in the sense of [KKZ4]? This question is still open, even for H = k G. One key idea in this paper is to relate the covariant ring with the so-called Hasse algebra (Definition 2.3) of the group G; this algebra has been considered before in the case that G is a Coxeter group, and was called the nilcoxeter algebra by Fomin-Stanley in [FS]. Asecondarygoalofthispaperistoprovesomerigidityresultsforgroupcoactions on some families of AS regular algebras. A remarkable rigidity theorem of Alev- Polo [AP, Theorem 1] states: Let g and g be two semisimple Lie algebras. Let G be a finite group of algebra automorphisms of U(g) such that U(g) G = U(g ). Then G is trivial and g = g. They also proved a rigidity theorem for the Weyl algebras [AP, Theorem 2]. The authors extended Alev-Polo s rigidity theorems to the graded case [KKZ1, Theorem 0.2 and Corollary 0.4]. For group coactions, we make the following definition. Definition 0.8. Let A be a connected graded algebra. We say that A is rigid with respect to group coaction if for every finite group G coacting on A homogeneously and inner faithfully, A kg is NOT isomorphic to A as algebras. Our main result concerning the rigidity of the dual group action is the following result, which provides a dual version of the rigidity results proved in [KKZ1, Theorem 0.2 and Corollary 0.4]. The proof is based on the structure results Theorems 0.3 and 0.5. Theorem 0.9. Let k be an algebraically closed field. The following AS regular algebras are rigid with respect to group coactions. (1) The homogenization of the universal enveloping algebra of a finite dimensional semisimple Lie algebra H(g). (2) The Rees ring of the Weyl algebra A n (k) with respect to the standard filtration.

5 DUAL REFLECTION GROUPS 5 (3) The non-pi Sklyanin algebras of global dimension at least 3. Remark By using ideas and results in [EW2], one can show that there is no nontrivial semisimple Hopf algebra actions on the algebras listed in Theorem 0.9. Combining [KKZ1, Theorem 0.2 and Corollary 0.4] with Theorem 0.9, one obtains that each algebra A in Theorem 0.9 is rigid with respect to semisimple Hopf algebra actions, or, equivalently, if H is a semisimple Hopf algebra acting on A inner faithfully and A H = A, then H = k. Based on the above remark, we have an immediate question: Question Let A be an algebra that is rigid with respect to finite group actions, see [KKZ1]. Is A rigid with respect to any semisimple Hopf algebra action? This paper is organized as follows. We provide background material on Artin- Schelter regular algebras, the Nakayama automorphism, and local cohomology in Section 1. We study the Hasse algebra and Poincaré polynomials of a finite group in Section 2. In Section 3, we prove some basic properties concerning dual reflection groups. The results about the Nakayama automorphisms are proved in Section 4, and the proofs of Theorems 0.3, 0.4, 0.5, and 0.6 appear at the end of Section 4. Theorem 0.9, about the rigidity of the dual group action, is proved in Section Preliminaries An algebra A is called connected graded if A = k A 1 A 2 and A i A j A i+j for all i,j N. The Hilbert series of A is defined to be H A (t) = i N(dim k A i )t i. The algebras that we use to replace commutative polynomial rings are the AS regular algebras [AS]. We recall the definition below. Definition 1.1. A connected graded algebra A is called Artin-Schelter Gorenstein (or AS Gorenstein, for short) if the following conditions hold: (a) A has injective dimension d < on the left and on the right, (b) Ext i A ( Ak, A A) = Ext i A (k A,A A ) = 0 for all i d, and (c) Ext d A( A k, A A) = Ext d A(k A,A A ) = k(l) for some integer l. Here l is called the AS index of A. If in addition, (d) A has finite global dimension, and (e) A has finite Gelfand-Kirillov dimension, then A is called Artin-Schelter regular (or AS regular, for short) of dimension d. Let M be an A-bimodule, and let µ,ν be algebra automorphisms of A. Then µ M ν denotes the induced A-bimodule such that µ M ν = M as a k-space, and where a m b = µ(a)mν(b) for all a,b A and m µ M ν (= M). Let 1 be the identity of A. We also use µ M (respectively, M ν ) for µ M 1 (respectively, 1 M ν ).

6 6 E. KIRKMAN, J. KUZMANOVICH AND J.J. ZHANG Let A be a connected graded finite dimensional algebra. We say A is a Frobenius algebra if there is a nondegenerate associative bilinear form, : A A k, which is graded of degree l. This is equivalent to the existence of an isomorphism A = A( l) as graded left (or right) A-modules. There is a (classical) graded Nakayama automorphism µ Aut(A) such that a,b = µ(b),a for all a,b A. Further, A = µ A 1 ( l) as graded A-bimodules. A connected graded AS Gorenstein algebra of injective dimension zero is exactly a connected graded Frobenius algebra. The Nakayama automorphism is also defined for certain classes of infinite dimensional algebras; see the next definition. Definition 1.2. Let A be an algebra over k, and let A e = A A op. (1) A is called skew Calabi-Yau (or skew CY, for short) if (a) A is homologically smooth, that is, A has a projective resolution in the category A e -Mod that has finite length and such that each term in the projective resolution is finitely generated, and (b) there is an integer d and an algebra automorphism µ of A such that { (E1.2.1) Ext i 0 i d A e(a,ae ) = 1 A µ i = d, as A-bimodules, where 1 denotes the identity map of A. (2) If (E1.2.1) holds for some algebra automorphism µ of A, then µ is called the Nakayama automorphism of A, and is usually denoted by µ A. (3) We call A Calabi-Yau (or CY, for short) if A is skew Calabi-Yau and µ A is inner (or equivalently, µ A can be chosen to be the identity map after changing the generator of the bimodule 1 A µ ). If A is connected graded, the above definition should be made in the category of graded modules and (E1.2.1) should be replaced by { (E1.2.2) Ext i 0 i d A e(a,ae ) = 1 A µ (l) i = d, where 1 A µ (l) is the shift of 1 A µ by degree l. We will use local cohomology later. Let A be a locally finite N-graded algebra and m be the graded ideal A 1. Let A-GrMod denote the category of Z-graded left A-modules. For each graded left A-module M, we define Γ m (M) = {x M A n x = 0 for some n 1 } = lim n Hom A(A/A n,m) and call this the m-torsion submodule of M. It is standard that the functor Γ m ( ) is a left exact functor from A-GrMod to itself. Since this category has enough injectives, the right derived functors R i Γ m are defined and called the local cohomology functors. Explicitly, one has R i Γ m (M) = lim n Exti A (A/A n,m). See [AZ, VdB] for more details. The Nakayama automorphism of a noetherian AS regular algebra can be recovered by using local cohomology [RRZ2, Lemma 3.5]: (E1.2.3) R d Γ m (A) = µ A 1 ( l)

7 DUAL REFLECTION GROUPS 7 where l is the AS index of A. 2. Poincaré polynomials and Hasse algebras Let G be a finite group, though some of the definitions make sense in the infinite case. Let e be the unit of G. We say R G is a set of generators of G, if R generates G and e R (R need not be a minimal set of generators of G). We recall some definitions. Definition 2.1. Let R be a set of generators of a group G. The length of an element g G with respect to R is defined to be l R (g) := min{n v 1 v n = g, for some v i R}. We define the length of the identity e to be 0. The length l R (g) is alsocalled reduced length by some authors. Definition 2.2. Let R be a set of generators of G. The Poincaré polynomial associated to R is defined to be p R (t) = g Gt l R(g). The notion of length of a group element with respect to R, where R is the set of Coxeter generators, is standard for Coxeter groups [BjB, p.15], and its generating function is called the Poincaré polynomial in [BjB, p. 201]. There is also interest in the generating function for other groups (e.g. the alternating group [Ro, p. 3] and [BRR, p. 849]); the article [He] gives a survey of recent progress in determining the order of magnitude of the maximal length of an element of a finite group G with respect to any generating set of G (i.e. the maximal degree of any Poincaré polynomial of the group) for many families of linear algebraic groups and permutation groups (often with respect to R R 1 ). If there is no confusion (for example, R is fixed), we might use p(t) instead of p R (t). Definition 2.3. Let R be a set of generators of a group G. (1) The Hasse algebra associated to R, denoted by H G (R), is the associated graded algebra of the group algebra kg with respect to the generating space ke+kr, H G (R) := (ke+kr) i /(ke+kr) i 1 i=0 where (ke+kr) 0 = k and (ke+kr) 1 = 0. (1 ) Equivalently, the Hasse algebra associated to R, denoted by H G (R), is the associative algebra with k-linear basis G together with multiplication determined by { gh l R (gh) = l R (g)+l R (h), g h = 0 l R (gh) < l R (g)+l R (h). (2) A skew Hasse algebra associated to R is an associative algebra with k-linear basis G together with multiplication determined by { α g,h gh l R (gh) = l R (g)+l R (h), g h = 0 l R (gh) < l R (g)+l R (h),

8 8 E. KIRKMAN, J. KUZMANOVICH AND J.J. ZHANG where {α g,h l R (gh) = l R (g)+l R (h)} is a set of nonzero scalars in k. Remark 2.4. (1) It is easy to see that Definitions 2.3(1) and 2.3(1 ) of the Hasse algebra are equivalent. (2) The first definition can be generalized to any finite dimensional Hopf algebra K as follows: Let V be a generating k-space of the associative algebra K with 1 V that is also a right coideal of K (namely, we have : V V K). The Hasse algebra associated to R is defined to be H K (V) := (k+v) i /(k+v) i 1. i=0 On the other hand, the second definition seems easier to understand in the group case. (3) Followingthedefinition, H G (R)isaconnectedgradedalgebrawithdeg(g) = l R (g) for all g G. (4) The Poincaré polynomial p R (t) is the Hilbert series of the graded algebra H G (R). (5) The Hasse algebra was called the nilcoxeter algebra by Fomin-Stanley [FS] for Coxeter group G with Coxeter generating set R. See [Al] for some related results. We begin with a few examples. Example 2.5. Let G be the quaternion group {±1,±i,±j,±k} with ij = k = ji, i 2 = j 2 = k 2 = 1. Let R be a generating set {x 1 := i,x 2 := j,x 3 := j}. Then elements of length two are {y 1 := k,y 2 := k,y 3 = 1} and the unique element of length 3 is z := i. The Poincarépolynomial is p R (t) = 1+3t+3t 2 +t 3 = (1+t) 3 and the Hasse algebra is H := H G (R) = k kx 1 kx 2 kx 3 ky 1 ky 2 ky 3 kz with multiplication determined as in Definition 2.3(1 ). Explicitly, we have multiplication x 1 x 2 x 3 y 1 y 2 y 3 z x 1 y 3 y 1 y z 0 x 2 y 2 y z 0 0 x 3 y 1 0 y 3 z y 1 0 z y z y 3 z z Itiseasytoseethatthisisanon-symmetricFrobeniusalgebraandtheNakayama automorphism of this algebra is determined by µ H : x 1 x 1, x 2 x 3, x 3 x 2 which has order 2. With some effort one can show that H is a Koszul algebra and its Koszul dual E is the AS regular algebra k a 1,a 2,a 3 /(a 1 a 2 + a 3 a 1,a 1 a 3 + a 2 a 1,a a2 2 + a2 3 ). Therefore the GK-dimension of the Ext-algebra of H is 3. Example 2.6. Let G be the symmetric group S 3 and R be the generating set {x 1 := (12),x 2 := (23)}. Then elementsoflength twoare{y 1 := (123),y 2 := (132)}

9 DUAL REFLECTION GROUPS 9 and the only element of length 3 is z := (13). The Poincaré polynomial is p R (t) = 1+2t+2t 2 +t 3 andthe Hassealgebrais H := H G (R) = k kx 1 kx 2 ky 1 ky 2 kz with multiplication determined by multiplication x 1 x 2 y 1 y 2 z x 1 0 y 1 0 z 0 x 2 y 2 0 z 0 0 y 1 z y 2 0 z z This is a non-symmetric Frobenius algebra and the Nakayama automorphism of this algebra is determined by the map of interchanging x 1 with x 2. It is easy to see that H is isomorphic to k a,b /(a 2,b 2,aba bab) which is not a Koszul algebra. The Ext-algebra E of the H, which was computed in [SV, Theorem 3.3], has GK-dimension 2. Example 2.7. Let G be the dihedral group D 2n with 2n elements, and let s 1 = r and s 2 = rρ be the usual Coxeter generators where r 2 = ρ n = e and ρr = rρ 1. The Poincaré polynomial of G with this choice of generating set is p R (t) = 1+2t+2t t n 1 +t n 1 = (1+t)(1+t+t 2 + +t n ). So the Hasse algebra has the following elements in each degree: degree 1: r,rρ; degree 2: ρ 1,ρ; degree 3: rρ 1,rρ 2 ; degree 4: ρ 2,ρ 2 ; etc. When nis odd the unique largestlength element is rρ n+1 2, while when n is evenit is ρ n 2, which is central. Hence, in the case when n is even, HD2n (R) is a symmetric Frobenius algebra, but a non-symmetric Frobenius algebra when n is odd. Based on the above examples, here are some natural questions. Question 2.8. (1) When is H G (R) a Frobenius algebra? (2) When is H G (R) a symmetric Frobenius algebra? (3) When is H G (R) a Koszul (or N-Koszul) algebra? (4) Let E G (R) be the Ext-algebra of the connected graded algebra H G (R). What is the GK-dimension of E G (R) in terms of (G,R)? We give an answer to the first question in the next theorem. A polynomial p(t) = n i=0 a it i of degree n is called palindrome if a i = a n i for all i. If H G (R) is Frobenius, then p R (t) is palindrome. As a consequence, the leading coefficient of p R (t) is 1. Theorem 2.9. Let R be a set of generators of G. Suppose that p R (t) is palindrome. Then the following hold. (1) H G (R) is Frobenius. (2) Let µ be the graded Nakayama automorphism of H G (R). Then µ is determined by a permutation of the elements in G of length 1. Proof. Since p R (t) is palindrome, the coefficient of the leading term of p R (t) is 1. This means that there is a unique element in G of the maximal length. Let be the multiplication of H G (R). The group product is suppressed.

10 10 E. KIRKMAN, J. KUZMANOVICH AND J.J. ZHANG Let m G be the unique element of the maximal length d. (Later we will see that m agrees with the mass element in Definition 0.2.) Let {x i1, x ini } be the complete list of elements in G of length i for 1 i d 1. We now prove the following claim: (a) For each x ij, there is a unique element of length s := d i, say x sw, such that x ij x sw = x ij x sw = m where x sw = x 1 ij m as in G. (b) If t w, then x ij x st = 0. (c) Every element of length d i is of the form x 1 ij m (for different j). Consequently, for any t w (where x sw is defined as in (a)), x ij x st = 0. We prove the claim by induction on i. Initial case i = 1: For each fixed j n 1, we consider x 1 1j m. Since x 1j e, x 1 1j m m. This means that l R(x 1 1j m) d 1. Since l R (m) = d and l R (x 1j ) = 1, l R (x 1 1j m) d 1. Thus l R(x 1 1j m) = d 1 and x1j 1 m = x d 1w for some w n d 1. We have x 1j x d 1w = x 1j x d 1w = m. For any t w, x 1j x d 1t m, so x 1j x d 1t = 0 by definition. Since p R (t) is palindrome, every element of length d 1 is of the form x 1 1j m. Inductive step: Let i 1. Suppose the claim holds for i i. Pick a x i+1j for any fixed j, then xi+1j 1 m is not equal to any x 1 i j m for all i i. Therefore x 1 i+1jm is not of the form x d i j for all i i and all j n d i. As a consequence, l R (x 1 i+1jm) d i 1. Since m = x i+1j (x 1 i+1j m), l R(x 1 i+1j m) d i 1. Thus l R(x 1 i+1jm) = d i 1. Write x 1 i+1j m as x d i 1w. For any t w, x i+1j x d i 1t m. By definition, x i+1j x d i 1t = 0. Since p R (t) is palindrome, every element of length d (i+1) is of the form x 1 (i+1)jm for some j. Therefore we proved the claim. (1) Recall that, if there is a nondegenerate associative bilinear form, : A A k and an algebra automorphism µ of A such that a,b = µ(b),a (= b,µ 1 (a) ) for all a,b A, then A is a Frobenius algebra and µ is a Nakayama automorphism of A. The Nakayama automorphism always exists for Frobenius algebras and the graded Nakayama automorphism is unique for a connected graded Frobenius algebra. Note that H G (R) is graded with m having the highest degree. Let pr km be the projection to the highest degree component of H G (R), and define a,b = pr km (ab). Then the claim implies that a, b is a nondegenerate associative bilinear form. Therefore H G (R) is Frobenius. (2) By the claim, there is a permutation σ S n1 such that (E2.9.1) x 1j x d 1w = m = x d 1w x 1σ(j) for all 1 j n 1. Using this observation we see that µ 1 maps x 1j to x 1σ(j). Since H G (R) is generated in degree 1, µ is completely determined by σ 1. The permutation given in Theorem 2.9(2) is called the Nakayama permutation of H G (R). In the setting of Theorem 2.9, there is the unique element in G of the maximal length with respect to the length function l. We give an answer to Question 2.8(2) next.

11 DUAL REFLECTION GROUPS 11 Theorem Suppose p R (t) is palindrome. Let m be the unique element of the maximal length. Let µ be the graded Nakayama automorphism of H G (R). (1) µ permutes the elements of G and preserves the length. (2) µ 1 equals the conjugation η m : g m 1 gm when applied to the basis element g G. (3) H G (R) is symmetric if and only if m is central. (4) Suppose p R (t) = 1+at+ i 2 a it i. Then m n is central for some n dividing a!. (5) The conjugation η m in part (2) preserves the length of g G. Proof. (1) By the proofoftheorem 2.9, µ 1 maps x ij to x ij wherej is determined by (E2.10.1) x ij x d iw = x ij x d iw = m = x d iw x ij = x d iw x ij. Hence the assertion follows. Since H G (R) is generated in degree 1, this automorphism µ 1 is completely determined by (E2.9.1). (2) By part (1) and (E2.10.1), for any (i,j), µ 1 (x ij ) = x ij = x 1 d iw m = m 1 (mx 1 d iw )m = m 1 x ij m = η m (x ij ) which implies the assertion. (3) Thisfollowsfromthe fact µ 1 (= η m ) isthe identityifandonlyifmiscentral. (4) Restricted to the degree 1 component of H G (R), η m is a permutation in S a. So η m has order n where n divides a!. This implies that η m n is the identity, which is equivalent to the assertion that m n is central. (5) This is an immediate consequence of parts (1,2). When G is a Coxeter group and (G,R) a Coxeter system, the Hasse algebra H G (R) agrees with the nilcoxeter algebra of Fomin-Stanley [FS], which was studied by several people; see, for example, [Al, Ba, KM]. The Frobenius property of the nilcoxeter algebras was proved by [Al] in which some statements in Theorems 2.9 and 2.10 are proven (for the special case of nilcoxeter algebras). There are other Frobenius algebras associated to (G, R), for example, the covariant algebra of (G,R) and the Nichols algebra (or the Nichols-Woronowicz algebra as it is called in [Ba, KM, MS]) B(V). Theorem If the Hasse algebra H G (R) is Frobenius, then so is any skew Hasse algebra associated to (G, R). Proof. Since H G (R) is Frobenius, p R (t) is palindrome. Then all statements made in the proof of Theorem 2.9 hold. Let A be a skew Hasse algebra associated to (G,R). Then A is a connected graded algebra with a basis {g g G}. Following the proof of Theorem 2.9, for each i, let {x i1, x ini } G be a k-linear basis of A i, which is the complete list of elements in G of length i. Then, by the definition of skew Hasse algebra, the following hold in the algebra A: (1) For each x ij, there is a unique element of length s := d i, say x sw, such that x ij x sw = x ij x sw = αm where x sw = x 1 ij m is in G and α is a nonzero scalar dependent on x ij and x sw. (2) If t w, then x ij x st = 0. (3) Every element of length d i is of the form x 1 ij m (for different j).

12 12 E. KIRKMAN, J. KUZMANOVICH AND J.J. ZHANG Let pr km be the projection from A to the highest degree component and define a,b = pr km (ab). It follows from (1,2,3) that a,b is a nondegenerate associative bilinear form. Therefore A is Frobenius. By arguments similar to those in the proofs of Theorems 2.9 and 2.10, we see that the graded Nakayama automorphism of A is of the form (E2.11.1) µ A : g β(g)mgm 1, g G where {β(g) g G} are nonzero scalars. 3. Dual reflection groups A commutative algebra is AS regular if and only if it is a polynomial ring, so we can extend the notion of a reflection group to groups acting on more general AS regular algebras using the following definition. Definition 3.1. A finite group G is called a reflection group (in the noncommutative setting) if kg acts homogeneously and inner faithfully on a noetherian AS regular domain A that is generated in degree 1 such that the fixed subring A G is AS regular. When A is a skew polynomial ring, a special AS regular algebra, reflections (or quasi-reflections as they were called in [KKZ1, Definition 2.2]) and reflection groups were studied in [KKZ1, KKZ2]. One interesting class of reflection groups that arises in this context are the so-called mystic reflection groups [KKZ2]. More generally, we can extend the definition of reflection group to semisimple Hopf algebras in the following definition. Definition 3.2. LetH be asemisimplehopfalgebra. We sayh isareflection Hopf algebra if H acts homogeneously and inner faithfully on a noetherian AS regular domain A that is generated in degree 1 such that the fixed subring A H is again AS regular. In this case we say that H acts on A as a reflection Hopf algebra. In [KKZ3, Examples 7.4 and 7.6] we show that H 8, the Kac-Palyutkin Hopf algebra of dimension 8 (a semisimple Hopf algebra that is not a isomorphic to a group algebra), is a reflection Hopf algebra for the skew polynomial algebras k i [u,v] and k 1 [u,v]. Further results on the structure and properties of reflection Hopf algebras are the object of research in progress. Comparing Definition 3.1 with Definition 3.2, it is clear that a reflection group is a special case of a reflection Hopf algebra. The main object of this paper is the dual reflection group in Definition 0.1, which is another spacial case of Definition 3.2. Let G be a finite group and let H be the dual Hopf algebra k G := (kg). We say G is a dual reflection group if H acts homogeneously and inner faithfully on a noetherian AS regular domain A that is generated in degree 1 such that the fixed subring A H is AS regular [Definition 0.1]. In this case we say G coacts on A as a dual reflection group. The group-theoretic or combinatorial connection between reflection groups and dual reflection groups is not yet evident; Example 3.7 is one example of a dual reflection group, and [KKZ5] will contain further examples. We note that the study of k G -actions on noncommutative algebras is an important and interesting topic related to different areas of mathematics even if G is not a dual reflection group and A is not an AS regular algebra [CM]. The algebraic and coalgebraic structure of k G were reviewed in introduction.

13 DUAL REFLECTION GROUPS 13 When A is G-graded, we can write A as g G A g. The following lemma is well-known. Lemma 3.3. Let A be a noetherian AS regular domain and H be a semisimple Hopf algebra acting homogeneously and inner faithfully on A. (1) [KKZ3, Lemma 2.4] The fixed subring A H is noetherian and A is finitely generated over A H on the left and on the right. (2) Suppose that the fixed subring A H is AS regular. Then gldima H = gldima and A is free over A H on both sides. Proof. (2) The statement of [KKZ1, Lemma 1.10(d)] is for H = kg, but the proof of [KKZ1, Lemma 1.10(d)] works for any semisimple Hopf algebra H. We need another general result. Lemma 3.4. Let A be a noetherian AS Gorenstein algebra and B be a subalgebra of A. Assume that Then (i) B is AS Gorenstein. (ii) A is finitely generated and free over B on both sides. (1) injdima = injdimb. Suppose further that (iii) AB 1 = B 1 A. Let C = A/I where I is the 2-sided ideal AB 1. Then (2) C is Frobenius. (3) l C = l A l B < 0. (4) µ(i) = I where µ is the Nakayama automorphism of A. Proof. By the faithful flatness, (ii) implies that B is noetherian. (1) Let µ and ν be the Nakayama automorphisms of A and B respectively. Let d A and d B be the injective dimensions of A and B, respectively (and l A and l B be the AS indices) as given in Definition 1.1. By [RRZ2, Lemma 3.5], µ A 1 [d A ]( l A ) and ν B 1 [d B ]( l B ) are rigid dualizing complexes over A and B, respectively. By [YZ, Theorem 3.2(i) and Proposition 3.9(i)] µ A 1 [d A ]( l A ) = RHom B (A, ν B 1 [d B ]( l B )) as (A,B)-bimodule complexes. Since A is free as a left B-module (and as a right B-module), the homology of RHom B (A, ν B 1 [d B ]( l B )) is concentrated in complex degree d B. As a consequence, d A = d B (that is part (1)) and (E3.4.1) µ A 1 ( l) = Hom B (A, ν B 1 ) where l := l A l B. Let C = A/B 1 A, which is a graded factor ring of A if B 1 A = AB 1. Let D = A/AB 1. Then A B k = D as left A-module and k B A = C as a right A-module.

14 14 E. KIRKMAN, J. KUZMANOVICH AND J.J. ZHANG Since A is a free left B-module, Hom B (A, ν B 1 ) is a free right B-module. Then we have quasi-isomorphisms of complexes of left A-modules, Hom B (A, ν B 1 ) B k = RHom B (A, ν B 1 ) L B k = RHom B (A, ν B 1 L B k) = RHom B (A,k) = Hom B (A,k) = Hom B (A,Hom k ( k k B,k)) = Hom k (k B A,k) = Hom k (C,k) = C, where the second = follows from the fact that B A is finitely generated free over B. (2,3,4) Now we assume that AB 1 = B 1 A =: I is a 2-sided ideal of A. In this case C = D. Since I is a 2-sided ideal of A, so is µ 1 (I). Then µ A 1 B k = µ (A/I) 1 = A/(µ 1 (I)) as left A-modules. By applying B k to (E3.4.1), the above computation shows that A/(µ 1 (I))( l) = µ A 1 ( l) B k = Hom B (A, ν B 1 ) B k = C as left A-modules. As a consequence, µ 1 (I) = I (which is part (4)) and (E3.4.2) C( l) = C as left C-modules. Now (E3.4.2) implies that C is Frobenius and l C = l = l A l B. Therefore parts (2,3) follow. When H = k G acts on A, then the fixed subring A H is the group graded e- component of A, denoted by A e. Theorem 3.5. Let A be a noetherian AS regular domain. In parts (3-5) we further assume that A is generated in degree 1. Suppose G coacts on A as a dual reflection group. Let H = k G. (1) There is a set of homogeneous elements {f g g G} A with f e = 1 such that A g = f g A e = A e f g for all g G. As a consequence, the nonzero component of A g with lowest degree has dimension 1. (2) I := A(A e ) 1 is a two-sided ideal and A/I = g G kf g. As a consequence, the covariant ring A cov H is Frobenius. (3) Suppose that, as an H-module, A 1 = g G (kp g ) ng where n g 0. If n g > 0 and g e, then n g = 1. (4) The set R := {g G n g > 0,g e} generates G. (5) A cov H is a skew Hasse algebra associated to the generating set R. As a consequence, A cov H is a Frobenius algebra generated in degree 1. Furthermore, the Hilbert series of A cov H is palindrome and is a product of cyclotomic polynomials. By part (1), A = g G A ef g, but f g is not invertible except for g = e. In some sense A is a twisted semi-group ring without the basis elements f g being invertible. Both part (2) and (5) ofthe abovetheoremprovethat A cov H is a Frobenius algebra generated in degree 1.

15 DUAL REFLECTION GROUPS 15 Proof of Theorem 3.5. (1)NotethatA = g G A g andeacha g isana e -bimodule. Since A is a domain and the k G -action on A is inner faithful, we obtain that A g 0 for all g G. By Theorem 3.3(2), A is free over A e. Hence each A g is free over A e. Let a be a nonzero element in A g 1, then aa g A e. Hence A g has rank one as a right A e -module, which implies that A g = f g A e for some homogeneous element f g. Therefore the lowest degree of the nonzero component of A g is degf g. By symmetry, A g = A e f g. The consequence is clear. (2) By part (1), f g A e = A e f g. Hence f g (A e ) 1 = (A e ) 1 f g. Therefore I = A(A e ) 1 = g G f g (A e ) 1 = g G(A e ) 1 f g = (A e ) 1 A which shows that I is a 2-sided ideal. It is clear that A/I = g G kf g. The consequence follows from Lemma 3.4(2) since A cov H = A/I. (3) If g e and n g > 0, then A g has lowest degree 1. By part (1), n g = 1. (4) Since A/I is generated in degree1, G is generated by {g G n g > 0,g e}. (5) The first assertion is easy to check and the second follows from Theorem Since A is free over A e, we have A = A e A cov H as vector spaces. Hence H A (t) = H A e(t)h A cov H(t). Since both H(A) 1 and H A e(t) 1 are products of cyclotomic polynomials, so is H A cov H(t). It is well-known that every product of cyclotomic polynomials is palindrome. Let R be any algebra and t be a normal nonzerodivisor element in R. The conjugation η t of R is defined to be η t (r) = t 1 rt for all r R. The inverse conjugation ηt 1 is denoted by φ t. By Theorem 3.5(1), A = f g B = g g G g GBf where B = A e is AS regular. Let η g and φ g be the graded algebra automorphisms of B defined by (E3.5.1) η g : x fg 1 xf g, and φ g := η f 1 : x f g g xfg 1 for all x B. Then we have f g x = φ g (x)f g for all x B. Define c g,h B such that (E3.5.2) for all g,h G. Then we have f g f h = c g,h f gh φ g φ h = φ fg φ fh = φ fgf h = φ cg,h φ gh where φ cg,h is defined by sending x c g,h xc 1 g,h. If l R(gh) = l R (g) + l R (h), then c g,h is a scalar and φ cg,h is the identity, whence φ g φ h = φ gh. Then associativity shows the following. Lemma 3.6. Retain the above notation. (1) c g,h is a normal element in B. (2) The following holds c g,h φ gh (b)c gh,k = φ g φ h (b)φ g (c h,k )c g,hk = φ cg,h φ gh (b)φ g (c h,k )c g,hk for all g,h,k G and b B.

16 16 E. KIRKMAN, J. KUZMANOVICH AND J.J. ZHANG Example 3.7. The dihedral group D 8 of order 8 is a dual reflection group. Note that G := D 8 is generated by r of order 2 and ρ of order 4 subject to the rρ = ρ 3 r. Let A be generated by x,y,z subject to the relations zx = qxz, yx = azy, yz = xy. for any a k and q 2 = 1. It is easy to see that A = k q [x,z][y;σ] where σ sends z to x and x to az. Hence A is an AS regular algebra of global dimension 3 that is not PI (for generic a). Define the G-degree of the generators of A as deg G (x) = r, deg G (y) = rρ, deg G (z) = rρ 2. It is easy to check that the defining relations of A are G-homogeneous. Therefore G coacts on A homogeneously and inner faithfully. Since r 2 = e, x 2,y 2,z 2 are in the fixed subring A e. By using the PBW basis {x i z j y k i,j,k 0} of A, one can easily check that x i z j y k A e if and only if all i,j,k are even. Another straightforward computation shows that A e = k[x 2,z 2 ][y 2,τ] where τ = σ 2 k[x 2,z 2 ]. Therefore A e is AS regular and G is a dual reflection group. Although other examples of dual reflection groups will appear in [KKZ5] we do not know another n with D 2n a dual reflection group. Example 3.8. Let G be the quaternion group of order 8. We claim that G is not a dual reflection group. Suppose to the contrary that G coacts on a noetherian AS regular algebra A generated in degree 1 as a dual reflection group. Then, by Theorem 3.5(5), A cov H is a Frobenius skew Hasse algebra, and its Hilbert series is a product of cyclotomic polynomials and palindrome. Any generating set R of G must contain two elements of order 4, so the degree of p R (t) is less than or equal to 3. Since p R (1) = 8 the only possibility is p R (t) = (1+t) 3 = 1+3t+3t 2 +t 3. Hence there must be 3 generators, say x 1,x 2,x 3, of G-degree not equal to e. Without loss of generality (and up to a conjugation and a permutation), the only possibility is x 1 = i,x 2 = j and x 3 = j. The Hasse algebra with respect to this R is given in Example 2.5. Let B = A kg. Then as a B-module we have A = B x 1 B x 2 B x 3 B x 1 x 2 B x 1 x 3 B x 2 1B x 3 1B deg G (f g ) : e i j j k k 1 i where the second line is the G-degree of the generator of each component. Since ji = i( j), we obtain a relation x 2 x 1 = ax 1 x 3 for some nonzero scalar a k. Similarly, the following relations are forced: for scalars b,c,d k. x 3 x 1 = bx 1 x 2, x 2 2 = cx2 1, x 2 3 = dx2 1,

17 DUAL REFLECTION GROUPS 17 We can rescale x 2 to take c = 1 (giving possibly new a,b,d) and rescale x 3 to maked = 1(possiblychanginga,b). Sincex 2 1 commuteswithx 2 andx 3,a 2 = b 2 = 1 and x 2 x 3 and x 3 x 2 commute with each other. Also x 2 3 = x 3x 2 1 = bx 1x 2 x 1 = abx 2 1 x 3 = abx 3 3 so ab = 1. Combining with a2 = b 2 = 1, we obtain that either a = b = 1 or a = b = 1. There could be other generators y j s of G-degree e and the other relations involving x i s and y j s, but we will get a contradiction using the subalgebra S of A generated by the x 1,x 2,x 3. We first claim that the Koszul dual of S is infinite dimensional. Using the comments about the scalars above, the relations in S are: x 2 x 1 ±x 1 x 3 = 0, x 3 x 1 ±x 1 x 2 = 0, x 2 2 x2 1 = 0, x 2 3 x2 1 = 0. Then Koszul dual (also called the quadratic dual) S! of S is generated by x 1, x 2, x 3 with relations: x 2 x 1 x 1 x 3 = 0, x 3 x 1 x 1 x 2 = 0, x x x 3 2 = 0, x 2 x 3 = 0, x 3 x 2 = 0. The quadratic algebra generated by x 1, x 2, x 3 subject to the first three relation is an AS regular algebra of global dimension 3, denoted by D. It is easy to see that x 2 x 3 +x 3 x 2,x 2 x 3 is a sequence of normal elements in D. Thus GKdimS! = GKdimD/(x 2 x 3 +x 3 x 2,x 2 x 3 ) = 1, which implies that S! is infinite dimensional. Next we consider the natural graded algebra map f : S A. Note that this map is injective when restricted to degree 1. By taking the Koszul dual, we have a graded algebra map f! : A! S!. Since f is injective in degree 1, f! is surjective in degree 1. Since S! is generated in degree 1, f! is surjective. We have shown that S! is infinite dimensional, so is A!. By [PP, Proposition 1.3.1, p. 7] A! is a subalgebra of Ext A(k,k). We obtain that Ext A (k,k) is infinite dimensional, a contradiction to the fact that A is AS regular. 4. Nakayama automorphisms For any algebra A, the Nakayama automorphism of A (if it exists) is denoted by µ A. In this section we study the interplay between the Nakayama automorphisms of the algebrash G (R), A, A#k G, A e and A cov kg. We need to use some facts about the local cohomology that were reviewed in Section Nakayama automorphism of H G (R) and skew Hasse algebras. By Theorem 2.10(2), we have (E4.0.1) µ HG(R) : g mgm 1, g G.

18 18 E. KIRKMAN, J. KUZMANOVICH AND J.J. ZHANG By (E2.11.1), the Nakayama automorphism of a skew Hasse algebra B is of the form (E4.0.2) µ B : g β(g)mgm 1, g G, where {β(g) g G} are nonzero scalars in k Nakayama automorphism of A e. Let φ g be defined as in (E3.5.1). Lemma 4.1. Let A be a noetherian AS Gorenstein algebra of injective dimension d, and let σ be a graded algebra automorphism of A. Let M be an A-bimodule. (1) R i Γ m (M σ ) = R i Γ m (M) σ for all i. (2) Hom k (M σ,k) = σ Hom k (M,k). (3) Suppose B is an AS Gorenstein subalgebra of A such that there is a G- graded decomposition A = g G f g B = g GB f g, for some elements {f g g G}, where each f g B(= B f g ) is isomorphic to 1 B φg. Then there is a Z G-graded isomorphism of B-bimodules µ (E4.1.1) A A 1 ( l A ) = R d Γ m (A) = fg { φgµb B 1 }( l B ) g G where f g is a B-central generator of bidegree ( degf g,g 1 ). (4) µ A maps B to B and Proof. (1,2) Straightforward. (3) By (E1.2.3), (E4.1.2) µ A B = φ m µ B or µ B = (η m µ A ) B. µ A A 1 ( l A ) = R d Γ m (A). Since A is finitely generated over B on both sides, R i Γ m (M) = R i Γ mb (M) for all i and all graded A-bimodules M [AZ]. Here m B = B 1. By Lemma 3.4(1), A and B have the same injective dimension, say d. Hence we have Z G-graded isomorphisms of B-bimodules R d Γ m (A) = R d Γ mb ( g G f g B) = g GR d Γ mb (f g 1B φg ) viewing f g as a B-central generator = g Gf g Rd Γ mb ( 1 B φg ) viewing f g as a B-central generator = g Gf g (Rd Γ mb (B) φg ) = g Gf g φg (R d Γ mb (B) ) = g Gf g φg ( µb B 1 )( l B ) = g Gf g µb φg B 1 ( l B ).

19 DUAL REFLECTION GROUPS 19 Combining the above with (E4.1.2), we obtain (E4.1.1). (4) Let s be the lowest Z-degree element in µa A 1 ( l A ), which corresponds to by (E4.1.1). For every b B, by using (E4.1.1), f m sµ A (b) = b s = s(µ B φ m )(b) for all b B. Then µ A (b) = µ(b)φ m (b) for all b B. The assertion follows Nakayama automorphism of A#k G. Since k G is commutative, µ k G is the identity. By [RRZ2, Theorem 0.2], the Nakayama automorphism of A#k G is given by µ A#k G = µ A #(µ k G Ξ l hdet) = µ A #Ξ l hdet where hdet is the homological determinant of the k G -action on A, and Ξ l hdet is the corresponding left winding automorphism. We start with a calculation of the hdet. Proposition 4.2. Retain the above notation. (1) The homological determinant hdet of the k G action on A is the projection of k G onto kp m 1 where m is the unique maximal length element in G. As a consequence, m 1 is the homological codeterminant of G-coaction on A. (2) l Ae = l A +degp R (t). (3) The left winding automorphism Ξ l hdet is the left translation transl m by m. Remark 4.3. Retain the notation as in Proposition 4.2. (1) Proposition 4.2(1) asserts that m is in fact the mass element of G-coaction on A defined in Definition 0.2. (2) Proposition 4.2(2) follows also from Lemma 3.4(3). Proof of Proposition 4.2. (1,2) Let H = k G. By [KKZ3, Proposition 5.3(d)], hdet is determined by the following: Let u be a nonzero element in Ext d A (k,a) where d is the injective dimension of A. Then there is an induced algebra homomorphism η : H k satisfying h u = η(h)u for all h H. The homological determinant hdet is equal to η S. Since H = k G, the η is the projection pr g from H to kp g for some g G. In this case, we just say that η corresponds to a group element g G. In fact, when we view A is a G-graded algebra, g is the G-degree of u. Thus hdet = η S = pr g 1 corresponds to the element g 1. Now we consider Ext d A e (k,a). Since A is free over A e on the left and the right, by the change of rings, there are isomorphisms of G-graded vector spaces Ext d A e (k,a) = Ext d A (A A e k,a) = Ext d A (Acov kg,a) = (A cov kg ) k Ext d A (k,a). Since A = g G f ga e = Ae A cov kg as G-graded A e -module, Ext d A e (k,a) = Ext d A e (k,a e ) A cov kg. By definition, Ext d A e (k,a e ) has bidegree ( l Ae,e), while Ext d A(k,A) has bidegree ( l A,g). Since Ext d A e (k,a) = Ext d A e (k,a e ) A cov kg, the lowest Z-degree element in Ext d A e (k,a) has bidegree (( l Ae,e), and the lowest Z-degree element in (A cov kg ) k Ext d A(k,A) has bidegree ( degp R (t), m) + ( l A,g). Therefore g = m and l Ae = l A +degp R (t). Hence the assertions follow.

20 20 E. KIRKMAN, J. KUZMANOVICH AND J.J. ZHANG (3) By definition, for any p h k G, Ξ l hdet(p h ) = hdet((p h ) 1 )(p h ) 2 = hdet(p s )p s 1 h = hdet(p m 1)p mh = p mh which is the left translation by m. We have an immediate consequence. Corollary 4.4. The Nakayama automorphism of the A#k G is given by µ A#k G = µ A #trans l m. Proof. Since k G is semisimple, S 2 is the identity. Then assertion follows from [RRZ2, Theorem 0.2] and Proposition Nakayama automorphism of A cov kg. We haveanexactsequenceofgraded algebras 1 A e A A cov kg 1. Also we can describe A cov kg as a skew Hasse algebra, with k-linear basis {f g g G}, and the multiplication of A cov kg is determined by { α(g,h)f gh l R (gh) = l R (g)+l R (h), f g f h = 0 l R (gh) < l R (g)+l R (h), where{α(g,h) l R (gh) = l R (g)+l R (h)}isasetofnonzeroscalarsink, seedefinition 2.3(2). Since l R (g) = l R (mgm 1 ) [Theorem 2.10] and l R (mg 1 ) = l R (m) l R (g) [Proof of Theorem 2.9], we have f mg 1 f g = α(mg 1,g)f m = α(mg 1,g)α(mgm 1,mg 1 ) 1 f mgm 1 f mg 1. Combining the above with (E2.11.1), the Nakayama automorphism of A cov kg is (E4.4.1) µ : f g β(g)f mgm 1 with (E4.4.2) β(g) = α(mg 1,g)α(mgm 1,mg 1 ) 1 for all g G. If f m is a normal element in A (which is a domain), then we can define the conjugation automorphism (E4.4.3) φ m : x f m xf 1 m, x A which agrees with the form given in (E3.5.1) when restricted to the subalgebra B = A cov kg. Proposition 4.5. Suppose G coacts on A as a dual reflection group. (1) f m is a normal element. (2) The Nakayama automorphism of A cov kg is equal to the induced automorphism of φ m of A given in (E4.4.3).

21 DUAL REFLECTION GROUPS 21 Proof. (1) Let B = A cov kg. If b B, then f m b = b f m by Theorem 3.5(1). If x = f g, we have f m f g = α(mgm 1,mg 1 ) 1 f mgm 1f mg 1f g = α(mgm 1,mg 1 ) 1 f mgm 1α(mg 1,g)f m = β(g)f mgm 1f m. Since A is generated by B and {f g g G}, f m is normal. (2) By the computation in the proof of part (1), f m f g = β(g)f mgm 1f m which is equivalent to φ m (f g ) = β(g)f mgm 1 for all g G. The assertion follows by (E4.4.1). We finish this section with proofs of the main results. Proof of Theorem 0.3. (1) This is Theorem 3.5(1). (2) This follows by combining parts (3) and (4) of Theorem 3.5. (3) By Theorem 3.5(5), A cov kg is a skew Hasse algebra. Note that deg(f g ) = deg(f g ) where f g is the image of f g in A cov kg. As a G-graded vector space, any skew Hasse algebra is isomorphic to the associated Hasse algebra. The assertion follows by Remark 2.4(3). (4) We use m to denote the element in G of the maximal length with respect to l R. By Remark 4.3(1), m equals the mass element defined in Definition 0.2. (5) Since A is free over A H and A = A H A cov H as graded vector spaces, p(t) is the Hilbert series of A cov H. Since A cov kg is a skew Hasse algebra, p(1) = G and degp(t) = l R (m). By Theorem 3.5(5), p(t) is palindrome and is a product of cyclotomic polynomials. Proof of Theorem 0.4. (1) By Lemma 3.4(2), C = A cov kg is Frobenius. Since A cov kg is a skew Hasse algebra [Theorem 3.5(5)], dima cov kg = G. (2) The first assertion is proven in the proof of Theorem 0.3(5). The consequence is clear. (3) This is (E4.4.1) or the proof of Proposition 4.5. (4) By Proposition 4.5(2), µ A cov k G is induced by the conjugation φ m : x f m xfm 1. Note that φ m(f m ) = f m, which implies that β(m) = 1. For any g,h G with l R (gh) = l R (g) + l R (h), we have f g f h = α(g,h)f gh for some nonzero scalar α(g, h). Then φ m (f gh ) = α(g,h) 1 φ m (f g f h ) = α(g,h) 1 φ m (f g )φ m (f h ) = α(g,h) 1 β(g)β(h)f mgm 1f mhm 1 = α(g,h) 1 β(g)β(h)α(mgm 1,mhm 1 )f mghm 1 = α(g,h) 1 α(mgm 1,mhm 1 )β(g)β(h)f mghm 1. Henceβ(gh) = α(g,h) 1 α(mgm 1,mhm 1 )β(g)β(h)forallg,hsatisfyingl R (gh) = l R (g)+l R (h). If m commutes with g and h, then β(gh) = β(g)β(h).